dirac cones - jdj.mit.edu

LETTERdoi:10.1038/nature14889

Spawning rings of exceptional points out ofDirac conesBo Zhen1*, Chia Wei Hsu1,2*, Yuichi Igarashi1,3*, Ling Lu1, Ido Kaminer1, Adi Pick1,4, Song-Liang Chua5,John D. Joannopoulos1 & Marin Soljacic1

The Dirac cone underlies many unique electronic properties ofgraphene1 and topological insulators, and its band structure—two conical bands touching at a single point—has also been rea-lized for photons in waveguide arrays2, atoms in optical lattices3,and through accidental degeneracy4,5. Deformation of the Diraccone often reveals intriguing properties; an example is thequantum Hall effect, where a constant magnetic field breaks theDirac cone into isolated Landau levels. A seemingly unrelated phe-nomenon is the exceptional point6,7, also known as the parity–timesymmetry breaking point8–11, where two resonances coincide inboth their positions and widths. Exceptional points lead tocounter-intuitive phenomena such as loss-induced transparency12,unidirectional transmission or reflection11,13,14, and lasers withreversed pump dependence15 or single-mode operation16,17. Diraccones and exceptional points are connected: it was theoreticallysuggested that certain non-Hermitian perturbations can deforma Dirac cone and spawn a ring of exceptional points18–20. Here weexperimentally demonstrate such an ‘exceptional ring’ in a photo-nic crystal slab. Angle-resolved reflection measurements of thephotonic crystal slab reveal that the peaks of reflectivity followthe conical band structure of a Dirac cone resulting from accidentaldegeneracy, whereas the complex eigenvalues of the system aredeformed into a two-dimensional flat band enclosed by an excep-tional ring. This deformation arises from the dissimilar radiationrates of dipole and quadrupole resonances, which play a role ana-logous to the loss and gain in parity–time symmetric systems. Ourresults indicate that the radiation existing in any open system canfundamentally alter its physical properties in ways previouslyexpected only in the presence of material loss and gain.

Closed and lossless physical systems are described by Hermitianoperators, which guarantee realness of the eigenvalues and a completeset of eigenfunctions that are orthogonal to each other. On the otherhand, systems with open boundaries7,21 or with material loss andgain9–17,19 are non-Hermitian6, and have non-orthogonal eigenfunc-tions with complex eigenvalues where the imaginary part correspondsto decay or growth. The most drastic difference between Hermitianand non-Hermitian systems is that the latter exhibit exceptional points(EPs) where both the real and the imaginary parts of the eigenvaluescoalesce. At an EP, two (or more) eigenfunctions collapse into one sothe eigenspace no longer forms a complete basis, and this eigenfunc-tion becomes orthogonal to itself under the unconjugated ‘inner prod-uct’6,7. To date, most studies of the EP and its intriguing consequencesconcern parity–time symmetric systems that rely on material loss andgain9–17,19, but EPs are a general property that require only non-Hermiticity. Here, we show the existence of EPs in a photonic crystalslab with negligible absorption loss and no artificial gain. When aDirac-cone system has dissimilar radiation rates, the band structureis altered abruptly to show branching features with a ring of EPs. We

provide a complete picture of this system, ranging from an analyticmodel and numerical simulations to experimental observations; takentogether, these results illustrate the role of radiation-induced non-Hermiticity that bridges the study of EPs and the study of Dirac cones.

We start by showing that non-Hermiticity from radiation candeform an accidental Dirac point into a ring of EPs. First, consider atwo-dimensional photonic crystal (Fig. 1a inset), where a square lattice(periodicity a) of circular air holes (radius r) is introduced in a dielec-tric material. This is a Hermitian system, as there is no material gain orloss and no open boundary for radiation. By tuning a system parameter(for example, r), one can achieve accidental degeneracy between aquadrupole mode and two degenerate dipole modes at the C point(centre of the Brillouin zone), leading to a linear Dirac dispersion dueto the anti-crossing between two bands with the same symmetry4,22.The accidental Dirac dispersion from the effective Hamiltonian model(see equation (1) below with cd 5 0) is shown as solid lines in Fig. 1a,agreeing with numerical simulation results (symbols). In the effectiveHamiltonian we do not consider the dispersionless third band (greyline) owing to symmetry arguments (Supplementary Information sec-tion I), although this third band cannot be neglected in certain calcula-tions, including the Berry phase and effective medium properties23.

Next, we consider a similar, but open, system: a photonic crystal slab(Fig. 1b inset) with finite thickness h. With the open boundary, modeswithin the radiation continuum become resonances because they radi-ate by coupling to extended plane waves in the surrounding medium.Non-Hermitian perturbations need to be included in the Hamiltonianto account for the radiation loss. To the leading order, radiation of thedipole mode can be described by adding an imaginary part 2icd to theHamiltonian, while the quadrupole mode does not radiate owing to itssymmetry mismatch with the plane waves24. Specifically, at theC pointthe system has C2 rotational symmetry (invariant under 180u rotationaround the z axis), and the quadrupole mode does not couple to theradiating plane wave because the former has a field profile E(r) that iseven under C2 rotation, E(r)~OC2 E(r), whereas the latter is odd,E(r)~OC2 E(r). The effective Hamiltonian is

Hef f ~v0 vg kj j

vg kj j v0icd

! "ð1Þ

with complex eigenvalues

v+~v0icd

2+vg

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikj j2k2

c

qð2Þ

where v0 is the frequency at accidental degeneracy, vg is the groupvelocity of the linear Dirac dispersion in the absence of radiation, jkj isthe magnitude of the in-plane wavevector (kx, ky), and kc:cd=2vg.Here, one of the three bands is decoupled from the other two andis not included in equation (1) (see Supplementary Information sec-tion II). In equation (2), a ring defined by jkj5 kc separates the k space

*These authors contributed equally to this work.

1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 2Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA.3Smart Energy Research Laboratories, NEC Corporation, 34 Miyuiga-ka, Tsukuba, Ibaraki 305-8501, Japan. 4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA. 5DSONational Laboratories, 20 Science Park Drive, Singapore 118230, Singapore.

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into two regions: inside the ring (jkj, kc), Re(v6) are dispersionlessand degenerate; outside the ring (jkj. kc), Im(v6) are dispersionlessand degenerate. In the vicinity of kc, Im(v6) and Re(v6) exhibitsquare-root dispersion (also known as branching behaviour6) insideand outside the ring, respectively. Exactly on the ring (jkj5 kc), thetwo eigenvalues v6 are degenerate in both real and imaginary parts;meanwhile, the matrix Heff becomes defective with an incompleteeigenspace spanned by only one eigenvector (1, 2i)T that is orthogonalto itself under the unconjugated ‘inner product’, given by aTb forvectors a and b. This self-orthogonality is the definition of EPs; hence,here we have not just one EP, but a continuous ring of EPs. We call it anexceptional ring.

Figure 1b, c shows the complex eigenvalues of the photonic crystalslab structure calculated numerically (symbols), which closely followthe analytic model of equation (2) shown as solid lines in the figure. InSupplementary Fig. 1, we show that the two eigenvectors indeedcoalesce into one at the EP, which is impossible in Hermitian systems(also see Supplementary Information section III). When the radius r ofthe holes is tuned away from accidental degeneracy, the exceptionalring and the associated branching behaviour disappear, as shown inSupplementary Fig. 2. Several properties of the photonic crystal slabcontribute to the existence of this exceptional ring. Owing to peri-odicity, one can probe the dispersion from two degrees of freedom,kx and ky, in just one structure. The open boundary provides radiationloss, and the C2 rotational symmetry differentiates the radiation loss ofthe dipole mode and of the quadrupole mode.

We can rigorously show that the exceptional ring exists in realisticphotonic crystal slabs, not just in the effective Hamiltonian model. Ourproof is based on the unique topological property of EPs: when thesystem parameters evolve adiabatically along a loop encircling an EP,the two eigenvalues switch their positions when the system returns toits initial parameters7,21,25, in contrast to the typical case where the twoeigenvalues return to themselves. Using this property, we numerically

show, in Supplementary Fig. 3 and Supplementary Information sec-tion IV, that the complex eigenvalues always switch their positionsalong every direction in the k space, and therefore prove the existenceof this exceptional ring. As opposed to the simplified effective Hamil-tonian model, in a real photonic crystal slab, the EP may exist at aslightly different magnitude of k and for a slightly different hole radiusr along different directions in the k space, but this variation is small andnegligible in practice (Supplementary Information section V).

To demonstrate the existence of the exceptional ring in such asystem, we fabricate large-area periodic patterns in a Si3N4 slab(n 5 2.02 in the visible spectrum, thickness 180 nm) on top of 6mmof silica (n 5 1.46) using interference photolithography24. Scanningelectron microscope (SEM) images of the sample are shown inFig. 2a, featuring a square lattice (periodicity a 5 336 nm) of cylin-drical air holes with radius 109 nm. We immerse the structure into anoptical liquid with a specified refractive index that can be tuned; acci-dental degeneracy in the Hermitian part is achieved when the liquidindex is selected to be n 5 1.48. We perform angle-resolved reflectivitymeasurements (set-up shown in Fig. 2b) between 0u and 2u along theCR X direction and the CR M direction, for both s and p polariza-tions. Details of the sample fabrication and the experimental setup canbe found in Supplementary Information section VI. The measuredreflectivity for the relevant polarization is plotted in the upper panelof Fig. 2c, showing good agreement with numerical simulation results(lower panel), with differences coming from scattering due to surfaceroughness, inhomogeneous broadening, and the uncertainty in themeasurements of system parameters. The complete experimentalresult for both polarizations is shown in Supplementary Fig. 4; thethird and dispersionless band shows up in the other polarization,decoupled from the two bands of interest.

The peaks of reflectivity (dark red colour in Fig. 2c) follow the linearDirac dispersion; this feature disappears for structures with differentradii that do not reach accidental degeneracy (experimental results in

Hermitian Non-Hermitian

a b c

0 0.01 0.010

kxa/2π kya/2π

Freq

uenc

y, ω

a/2π

c

0.01 0 0.01|k|a/2π

Γ XM

0.01 0 0.01|k|a/2π

–2

–1

0Γ XMΓ

0.60

0.59

Freq

uenc

y, R

e(ω

)a/2πc

Im(ω

)a/2πc

× 1

03Im

(ω)a

/2πc

× 1

03

0.01 0|k|a/2π

M

0.60

0.59

0

–3

2rc

h

X

0.01

2r0.54

0.55

–0.01Freq

uenc

y, ω

a/2π

c

0.54

0.55

–0.01 Freq

uenc

y, R

e(ω

)a/2πc

0 0.01 0.010

kxa/2π kya/2π

–0.01–0.01

0 0.01 0.010

kxa/2π kya/2π

–0.01–0.01

–2

–1

Figure 1 | Accidental degeneracy in Hermitian and non-Hermitianphotonic crystals. a, Band structure of a two-dimensional photonic crystalconsisting of a square lattice of circular air holes. Tuning the radius r leads toaccidental degeneracy between a quadrupole band and two doubly degeneratedipole bands, resulting in two bands with linear Dirac dispersion (red and blue)and a flat band (grey). b, c, The real (b) and imaginary (c) parts of theeigenvalues of an open, and therefore non-Hermitian, system: aphotonic crystal slab with finite thickness, h. By tuning the radius, accidentaldegeneracy in the real part can be achieved, but the Dirac dispersion isdeformed owing to the non-Hermiticity. The analytic model predicts that the

real (imaginary) part of the eigenvalue stays as a constant inside (outside) a ringin the wavevector space, indicating two flat bands in dispersion, with a ring ofexceptional points (EPs) where both the real and the imaginary parts aredegenerate. The orange shaded regions correspond to the inside of the ring. Inthe upper panels of a–c, solid lines are predictions from the analytic model andsymbols are from numerical simulations: red squares represent the bandconnecting to the quadrupole mode at the centre; blue circles represent theband connecting to the dipole mode at the centre; and grey crosses represent thethird band that is decoupled from the previous two due to symmetry. The three-dimensional plots in the lower panels are from simulations.

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G2015 Macmillan Publishers Limited. All rights reserved

Supplementary Fig. 5). Note that the reflection peaks do not follow thereal part of the complex eigenvalues of the Hamiltonian; in fact theyfollow the eigenvalues of the Hermitian part of the Hamiltonian, eventhough the Hamiltonian is non-Hermitian. To understand this, weconsider a more general two-by-two Hamiltonian of a coupled res-onance system H and separate it into a Hermitian part A and an anti-Hermitian part 2iB (A and B are both Hermitian)

H~v1 k

k v2

! "

|fflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflA

ic1 c12

c12 c2

! "

|fflfflfflfflfflfflfflfflfflzfflfflfflfflfflfflfflfflffliB

######?eigenvalues vz 0

0 v

! "ð3Þ

As before, we use v6 to denote the complex eigenvalues of theHamiltonian A 2 iB. Physically, matrix A describes a lossless system,and matrix 2iB adds the effects of loss. In B, the diagonal elements areloss rates (in our system, they come primarily from radiation), and theoff-diagonal elements arise from overlap of the two radiation patterns,also known as external coupling of resonances via the continuum.Modelling the reflectivity using temporal coupled-mode theory(TCMT), we show that when matrix B is dominated by radiation,the reflection peaks occur near the eigenvalues V1,2 of the Hermitianpart A and are independent of the anti-Hermitian part 2iB (seeSupplementary Information section VII and Supplementary Fig. 6for details). Therefore, the linear Dirac dispersion observed in themeasured data of Fig. 2c (dark red) indicates that we have successfullyachieved accidental degeneracy in the eigenvalues of the Hermitianpart, consistent with the simplified model in equation (1). InSupplementary Fig. 8b, we plot the values of V1,2 extracted from the

reflectivity data through a more rigorous data analysis using TCMT(described below); the linear dispersion is indeed observed. We notethat when there is substantial non-radiative loss or material gain in thesystem, the reflection peaks no longer follow the eigenvalues of theHermitian part (see Supplementary Information section VIII andSupplementary Fig. 7).

The real part of the complex eigenvalues of the Hamiltonian,Re(v6), behave very differently from the reflectivity peaks.Simulation results (solid white lines in the lower panel of Fig. 2c) showRe(v6) is dispersionless at small angles with a branch-point singular-ity around 0.31u—consistent with the feature predicted by the simpli-fied Hamiltonian in equation (2). In Fig. 2d, we compare thereflectivity spectra from simulations (with peaks indicated by redarrows) with the corresponding complex eigenvalues at three repres-entative angles (0.8u in blue, 0.31u in green and 0.1u in magenta). At0.31u, the two complex eigenvalues are degenerate, indicating an EP;however, the two reflection peaks do not coincide since they representthe eigenvalues of only the Hermitian part of the Hamiltonian, whichdoes not have degeneracy here. The dip in reflectivity between the twopeaks (marked as black arrows in Figs 2 and 3) is the coupled-res-onator-induced transparency (CRIT) that arises from the interferencebetween radiation of the two resonances26, similar to electromagnet-ically induced transparency.

Qualitatively, the peak locations of the measured reflectivity spec-trum reveal the eigenvalues of the Hermitian part, A, and the line-widths of the peaks reveal the anti-Hermitian part, 2iB; diagonalizingA 2 iB yields the eigenvalues v6, as illustrated in equation (3). Tobe more quantitative, we use TCMT and account for both the direct

Simulation

Experiment

cΓM X

s-polarizedp-polarized d

Frequency, ωa/2πc

1Angle (degrees)

1 0 10.5 0.5

a

2r

h = 180nm

b

Source

Polarizer

SP

Sample

Mirror

BS

2Liquid

0.1 0.31 0.8W

avel

engt

h (n

m)

568

566

564

562

560

0

1 = 0.1°

–2

1 = 0.8°

0

–2

0.5

0

1

–2

= 0.31°

0.5

0.5

0

0.6

0.598

0.596

0.594

0.592

=

568

566

564

562

560W

avel

engt

h (n

m)

Frequency, ωa/2πc

0.6

0.598

0.596

0.594

0.592

0.592 0.601

CRIT

Im(ω

)(2πc

/a ×

10–3

)R

Im( ω

)(2πc

/a ×

10–3

)R

Im( ω

)(2πc

/a ×

10–3

)R

0.592 0.601

xy

Γ X

M

EP

0.592 0.601

a = 336nm

Re(ω)(2πc/a)

Re(ω)(2πc/a)

Re(ω)(2πc/a)

Figure 2 | Experimental reflectivity spectrum and accidental Diracdispersion. a, SEM images of the photonic crystal samples: side view (upperpanel) and top view (lower panel). b, Schematic drawing of the measurementset-up. Linearly polarized light from a super-continuum source is reflectedoff the photonic crystal slab (‘sample’) immersed in an optical liquid, andcollected by a spectrometer (SP). The incident angle h is controlled using aprecision rotating stage. BS, beam splitter. c, Reflectivity spectrum of the samplemeasured experimentally (upper panel) and calculated numerically (lowerpanel) along the CR X and the CR M directions. The peak locationof reflectivity reveals the Hermitian part of the system, which forms Diracdispersion due to accidental degeneracy. In the lower panel, white solid lines

indicate the real part of the eigenvalues; spectra and eigenvalues at threerepresentative angles (marked by dashed lines and circles) are shown ind. d, Three line cuts of reflectivity R from simulation results. Also shown are thecomplex eigenvalues (open circles) calculated numerically. At large angles(0.8u), the two resonances are far apart, so the reflectivity peaks (red arrows) areclose to the actual positions of the complex eigenvalues. However, at smallangles (0.3u, 0.1u), the coupling between resonances cause the resonance peaks(red arrows) to have much greater separations in frequency compared to thecomplex eigenvalues. The black arrows mark the dips in reflectivity thatcorrespond to the coupled-resonator induced transparency (CRIT, see textfor details).

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LETTER RESEARCH


(non-resonant) and the resonant reflection processes including nearbyresonances; the expression for reflectivity is given in SupplementaryInformation equation (S20), with the full derivation given inSupplementary Information section IX. Fitting the reflectivity curveswith the TCMT expression gives us an accurate estimate of the matrixelements and the eigenvalues; this procedure is the same as ourapproach in ref. 27 except that here we additionally account for thecoupling between resonances28. Figure 3a compares the fitted and themeasured reflectivity curves at three representative angles (with morecomparison in Supplementary Fig. 8a); the excellent agreement showsthe validity of the TCMT model. Underneath the reflectivity curves, weshow the complex eigenvalues. The difference between numericallycalculated reflectivity (Fig. 2d) and experimental results (Fig. 3a) stemsfrom the non-radiative decay channels in our system, mostly due toscattering loss from the surface roughness24.

Repeating the fitting procedure for the reflectivity spectrum mea-sured at different angles, we obtain the dispersion curves for all com-plex eigenvalues, which are plotted in Fig. 3b. Along both directions ink space (CR X and CR M), the two bands of interest (shown inblue and red) exhibit the EP behaviour predicted in equation (2): forjkj, kc the real parts are degenerate and dispersionless; for jkj. kc theimaginary parts are degenerate and dispersionless; for jkj in the vicin-ity of kc branching features are observed in the real or imaginary part.In Fig. 3c, we plot the eigenvalues on the complex plane for both theCR X and CR M directions. We can see that in both directions, thetwo eigenvalues approach each other and become very close at a cer-tain k point, which is a clear signature of the system being very close toan EP.

We have shown that non-Hermiticity arising from radiation cansignificantly alter fundamental properties of the system, including theband structures and the density of states; this effect becomes most

prominent near EPs. The photonic crystal slab described here providesa simple-to-realize platform for studying the influence of EPs on light–matter interaction, such as for single particle detection21 and modu-lation of quantum noise. The two-dimensional flat band can alsoprovide a high density of states and therefore high Purcell factors.The strong dispersion of loss in the vicinity of theC point can improvethe performance of large-area single-mode photonic crystal lasers29.The deformation into an exceptional ring is a general phenomenonthat can also be achieved with material gain or loss and for Dirac pointsin other lattices19,20. Further studies could advance the understandingof the connection between the topological property of Dirac points30

and that of EPs25 in general non-Hermitian wave systems, and ourmethod could go beyond photons to phonons, electrons and atoms.

Received 2 April; accepted 29 June 2015.Published online 9 September 2015.

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b

Angle (degrees)1 0 10.5 0.5

560

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564

566

Wavelength (nm

)

ΓM X0.6

0.598

0.596

0.594

Angle (degrees)1 0 10.5 0.5

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/a ×

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Im(ω

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/a ×

10–3

)

Im(ω

)a/2πc

× 1

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RIm

(ω)

(2πc

/a ×

10–3

)R

Im(ω

)a/2πc × 103

Freq

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)a/2πc

Re(ω)a/2πc Re(ω)a/2πc

Re(ω)(2πc/a)

Re(ω)(2πc/a)

Re(ω)(2πc/a)

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imaginary part (right panel). Red squares and dashed lines are used for the bandwith zero radiation loss at the C point, blue circles and dashed lines for theband with finite radiation loss at the C point, and grey crosses and dashed linesfor the third band decoupled from the previous two owing to symmetry.The orange shaded regions correspond to the inside of the ring. c, Positions ofthe eigenvalues (red and blue dashed lines) approach and become very close toeach other (indicated by the two brown arrows), demonstrating near-EPfeatures in different directions in the momentum space and the existence ofan exceptional ring.

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Supplementary Information is available in the online version of the paper.

Acknowledgements We thank T. Savas for fabrication of the samples, and F. Wang,Y. Yang, N. Rivera, S. Skirlo, O. Miller and S. G. Johnson for discussions. This work waspartly supported by the Army Research Office through the Institute for SoldierNanotechnologies under contract nos W911NF-07-D0004 and W911NF-13-D-0001.B.Z., L.L. and M.S. were partly supported by S3TEC, an Energy Frontier Research Centerfunded by the US Department of Energy under grant no. DE-SC0001299. L.L. wassupported in part by the Materials Research Science and Engineering Center of theNational Science Foundation (award no. DMR-1419807). I.K. was supported in part byMarie Curie grant no. 328853-MC-BSiCS.

Author Contributions All authors discussed the results and made critical contributionsto the work.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of the paper. Correspondenceand requests for materials should be addressed to B.Z. ([email protected]) and C.W.H.([email protected]).

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LETTER RESEARCH



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Section I. Effective Hamiltonian of accidental Dirac points in Hermitian systemsHere we consider the Hermitian system of a 2D square-lattice PhC tuned to accidental de-generacy between a quadrupole mode and two degenerate dipole modes at the Brillouin zonecenter (Γ point). Near Γ point where the in-plane wavevectors kx and ky are small, the effectiveHamiltonian given by first-order degenerate perturbation theory is [1]

H2Deff =

⎛

⎝ω0 vgkx vgkyvgkx ω0 0vgky 0 ω0

⎞

⎠ , (S.1)

which can be transformed to

UH2Deff U

−1 =

⎛

⎝ω0 vg|k| 0

vg|k| ω0 00 0 ω0

⎞

⎠ (S.2)

with the orthogonal rotation matrix

U =

⎛

⎝1 0 00 cos θ sin θ0 − sin θ cos θ

⎞

⎠ (S.3)

where cos θ = kx/|k|, sin θ = ky/|k|, and |k| =√k2x + k2

y . After transformation, the 3 × 3matrix becomes two isolated blocks: the upper 2 × 2 block gives the conical Dirac dispersion(ω = ω0 ± vg|k|), while the lower block is the intersecting flat band (ω = ω0).

Section II. Effective non-Hermitian Hamiltonian of the exceptional ringFor a 3D PhC slab with finite thickness, the two dipole modes become resonances with finitelifetime due to their radiation, with eigenvalues becoming complex (ω0 − iγd). With C4 rota-tional symmetry, these two dipole modes are related by a 90 rotation and therefore share thesame complex eigenvalue. Meanwhile, the quadrupole mode does not couple to radiation at theΓ point due to symmetry mismatch [2], and to leading order its eigenvalue remains at ω0. Theeffective non-Hermitian Hamiltonian of the 3D PhC slab thus becomes

H3Deff =

⎛

⎝ω0 vgkx vgkyvgkx ω0 − iγd 0vgky 0 ω0 − iγd

⎞

⎠ , (S.4)

which transforms to

UH3Deff U

−1 =

⎛

⎝ω0 vg|k| 0

vg|k| ω0 − iγd 00 0 ω0 − iγd

⎞

⎠ (S.5)

with the same matrix U as in equation (S.3). The upper 2 × 2 block is the Heff in equation (1)of the main text that gives rise to an exceptional ring, while the lower block is the flat band.

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Section III, Eigenfunctions in Hermitian and non-Hermitian systemsIn this section, we show the numerically calculated eigenfunctions in the Hermitian PhC andthe non-Hermitian PhC slab, and compare them to the analytical predictions from first-orderperturbation theory.

According to equation S.1, the eigenfunctions of the Hermitian PhC along the kx axis areΨ 1 ±Ψ2 with eigenvalues ω0 ± vgkx. Fig. S1a shows that the numerical results agree with thisprediction. These two eigenfunctions are orthogonal to each other, as expected for Hermitiansystems.

In comparison, for the non-Hermitian PhC slab, equation S.4 predicts the two eigenfunc-tions to merge into one, Ψ1 − iΨ2, at the exceptional point (EP). This prediction also agreeswith our numerical results, shown in Fig. S1b. Both eigenfunctions become very close to theanalytical prediction of Ψ1 − iΨ2. Furthermore, this coalesced eigenfunction is self-orthogonalunder the unconjugated product. Such coalescence of eigenfunctions can never happen in Her-mitian systems and highlights the effect of non-Hermiticity.

Section IV, Existence of exceptional points along every direction in momentum spaceIn this section, we demonstrate that EPs exist in all directions in the k space, not only for asimplified Hamiltonian (equation (1)), but also for realistic structures. To prove their existence,we use the unique topological property of EPs: when the system evolves adiabatically in theparameter space around an EP, the eigenvalues will switch their positions at the end of theloop [3,4]. In our system, the parameter space in which we choose to evolve the eigenfunctionsis three-dimensional, consisting of the two in-plane wavevectors (kx, ky) and the radius of theair holes r, as shown in Fig. S3a. Here, r can also be other parameters, like the refractive indexof the PhC slab (n), the periodicity of the square lattice (a), or the thickness of the slab (h). Forsimplicity of this demonstration, we choose r as the varying parameter while keeping all otherparameters (n, a, and h) fixed throughout.

First, we compare the evolution of the eigenvalues when the system parameters follow (1)a loop that does not enclose an EP, and (2) one that encloses an EP. Following the loop A →B → C → D → A in Fig. S3a,b that does not enclose an EP (point EP), we see that thecomplex eigenvalues come back to themselves at the end of the loop (Fig. S3c where the reddot and the blue dot return to their initial positions at the end of the loop). However, followingthe loop A′ → B′ → C ′ → D′ → A′ in Fig. S3d, which encloses an EP (point EP), we see thatthe complex eigenvalues switch their positions in the complex plane (Fig. S3f where the red dotand the blue dot switch their positions). This switching of the eigenvalues shows the existenceof an EP along the Γ to X direction, at some particular value of kx and some particular value ofradius r. This shows the existence of the EP without having to locate the exact parameters ofkx and r at which the EP occurs.

Similarly, we can evolve the parameters along any direction θ = tan(ky/kx) in the k spaceand check whether an EP exists along this direction or not. As two examples, we show the evo-lution of the complex eigenvalues when we evolve the parameters along the θ = π/8 directionfollowing the loop A′′ → B′′ → C ′′ → D′′ → A′′ and along the θ = π/4 direction following


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the loop A′′′ → B′′′ → C ′′′ → D′′′ → A′′′ in Fig. S3g,h. In both cases, we observe the switch-ing of the eigenvalues, showing the existence of an EP along these two directions. The sameshould hold for every direction in k space.

The above calculations show that for every direction θ we examined in the k space, there isalways a particular combination of kc and rc, which supports an EP. However, we note that ingeneral, different directions can have different kc and different rc, so the exceptional ring for therealistic PhC slab structure is parameterized by kc(θ) and rc(θ). This angular variation of kc(θ)and rc(θ) can be described by introducing higher order corrections in the effective Hamiltonian,which we examine in the next section.

Section V, Generalization of the effective HamiltonianHere, we generalize the effective Hamiltonian in equations (1) and (S.5). First, the radiation ofthe quadrapole mode is zero only at the Γ point; away from the Γ point, the quadrapole modehas a k-dependent radiation that is small but non-zero, which we denote with γq. Second, weconsider possible deviation from accidental degeneracy in the Hermitian part, with a frequencywalk-off δ. With these two additional ingredients, the effective Hamiltonian becomes

(ω0 + δ vgkvgk ω0

)− i

(γq

√γqγd√

γqγd γd

), (S.6)

with complex eigenvalues of

ω± = ω0 +δ

2− i

γq + γd2

±

√(vgk − i

√γqγd

)2 −(γd − γq

2− i

δ

2

)2

, (S.7)

which are generalized forms of equations (1) and (2), respectively. We note that the off-diagonalterm √

γqγd in equation (S.6) is required by energy conservation and time-reversal symme-try [5], as we will discuss more in the last section. Equation (S.7) shows that EP occurs whenthe two conditions

k = (γd − γq)/(2vg) ≈ γd/2vg,

δ = 2√γdγq,

(S.8)

are satisfied. In the region of momentum space of interest, γq is much smaller than γ0 (this canbe seen, for example, from the imaginary parts of Fig. S2a,c), so the first condition becomeskc ≈ γd/2vg, same as in the simplified model. For a given direction θ in the k space (asdiscussed in the previous section), we can vary the magnitude k and the radius r to find thekc(θ) and rc(θ) where these two conditions are met simultaneously.

We can now analyze the angular dependence of kc(θ) and rc(θ) without having to find theirexact values. The first condition of equation (S.8) says that the angular dependence of kc(θ)comes from γd and vg; in the PhC slab structure here, we find that γd varies by about 20%as the angle θ = tan(ky/kx) is varied, while vg remains almost the same; therefore, kc(θ)potentially varies by around 10% along the exceptional ring. For the second condition of equa-tion (S.8), we have γd ≈ 5 × 10−3ω0 and γq ≈ 5 × 10−5ω0 for our PhC slab structure, so



δc = 2√γdγq ≈ 1 × 10−3ω0. Again, γd and γq vary by about 20% as the angle θ is changed,

so δc can change by around 2× 10−4ω0. Empirically, we find that a change of δ by 2× 10−4ω0

corresponds to a change in the radius r of around 0.06 nm, which is the estimated range ofvariation for rc(θ) of all θ ∈ [0, 2π). This angular variation is much smaller than our structurecan resolve in practice, since the radii of different holes within one fabricated PhC slab willalready differ by more than 0.06 nm. So, in practice a given fabricated structure can be close toEP along all different directions θ, but is unlikely to be an exact EP for any direction.

Section VI, Sample fabrication and experimental detailsThe Si3N4 layer was grown with the low-pressure chemical vapor deposition method on a 6µm-thick cladding of SiO2 on the backbone of a silicon wafer (LioniX). Before exposure, the waferwas coated with a layer of polymer as anti-reflection coating, a thin layer of SiO2 as an inter-mediate layer for etching, and a layer of negative photoresist for exposure. The square latticepattern was created with Mach−Zehnder interference lithography using a 325-nm He/Cd laser.The angle between the two arms of the laser beam was chosen for a periodicity of 336 nm. Afterexposures, the pattern in the photoresist was transferred to Si3N4 by reactive-ion etching.

The source was a supercontinuum laser from NKT Photonics (SuperKCompact). A polar-izer selected s- or p-polarized light. The sample was immersed in a colorless liquid with tunablerefractive indices (Cargille Labs). The sample was mounted on two perpendicular motorizedrotation stages (Newport): one to orient the PhC to the Γ-X or Γ-M direction, and the other todetermine the incident angle θ. The reflectivity spectra were measured with a spectrometer withspectral resolution of 0.02 nm (HR4000; Ocean Optics).

Section VII, Model of reflection from two resonancesIn Section IX, we will derive the temporal-coupled coupled theory (TCMT) [5–10] equationsdescribing the full experimental system. Here, we first consider a simpler scenario to illustratethe qualitative features of the reflectivity spectrum. Consider two resonances in a photoniccrystal slab, governed by the Hamiltonian in equation (3) of the main text,

H =

(ω1 κκ ω2

)− i

(γ1 γ12γ12 γ2

), (S.9)

where the first term is the Hermitian part describing a lossless system, and the second term isthe anti-Hermitian part describing radiation of the two resonances. We ignore non-radiativeloss here for simplicity. Also, we ignore the direct Fresnel reflection between the dielectriclayers. We consider the photonic crystal slab to be mirror symmetric in z direction, with thetwo resonances having the same mirror symmetry in z. This simple system describes the basicelements of resonant reflection. TCMT predicts the reflectivity to be (see Section IX for the fulldetails)

R(ω) =1

1 + f 2(ω), f(ω) =

(ω − ω1)(ω − ω2)− κ2

γ1(ω − ω2) + γ2(ω − ω1) + 2γ12κ(S.10)



with the transmission being T (ω) = 1 − R(ω) since there is no non-radiative loss. We im-mediately see that the reflectivity reaches its maximal value of 1 when the numerator of f(ω)vanishes, which happens at two frequencies ω = Ω1,2 with

Ω1,2 =1

2

[ω1 + ω2 ±

√(ω1 − ω2)2 + 4κ2

](S.11)

being the eigenvalues for the Hermitian part of the Hamiltonian. Note that the anti-Hermitianpart determines the linewidth but has no effect on the location of the reflectivity peaks. This isthe first main conclusion of this section.

Secondly, we also observe that the reflectivity reaches its minimal value of 0 when the de-nominator of f(ω) vanishes, which happens at one frequency ω = (γ1ω2 + γ2ω1−2γ12κ)/(γ1 + γ2)that is in between Ω1 and Ω2. This is called coupled-resonator-induced transparency (CRIT) [11,12].

We emphasize that the reflectivity peaks, following eigenvalues of the Hermitian part of theHamiltonian, are different from the real part of the complex eigenvalues of the HamiltonianH . Consider a simple example with κ = 0 (so that Ω1,2 = ω1,2), Ω1,2 = ω0 ± b, and γ1 =γ2 = γ12 = b. The reflection peaks at Ω1,2 = ω0 ± b, while the two complex eigenvalues aredegenerate at ω+ = ω− = ω0 − ib, whose real part is in the middle of the two reflection peaks.This explains the reflectivity from the PhC slabs at 0.3 shown in Fig. 2d and Fig. 3a, where thedegenerate complex eigenvalues of the system are in between the two reflection peaks.

In Fig. S6, we use some examples to illustrate the difference between the reflectivity peaksand the real part of the complex eigenvalues. Fig. S6a shows the case when there is only oneresonance with complex eigenvalue ω0 − iγ; in this case, the reflection peak (red arrow) is atthe same position as the real part of the complex eigenvalue. In contrast, Fig. S6b shows thecase when there are two resonances [equation (S.10)] with κ = 0 and Ω1,2 fixed at ω0 ± b; aswe vary γ1,2, the complex eigenvalues (circles) vary accordingly, whereas the reflectivity peaks(red arrows) always show up at Ω1,2.

Section VIII, Effects of material loss and gainThe conclusion that reflectivity peaks at the eigenfrequencies Ω1,2 of the Hermitian part is validwhen radiation is the only source of loss and when there is no gain in the system. It no longerholds when substantial material loss and/or gain is introduced, as we show in this section.

To account for material loss and gain, we extend the Hamiltonian in Section VII to

H =

(ω1 κκ ω2

)− i

(γ1 γ12γ12 γ2

)

︸︷︷︸Radiation loss

− i

(η1 00 η2

)

︸︷︷︸Material loss/gain

(S.12)

with positive (negative) values of η1,2 describing material loss (gain). Same derivation using



TCMT yields reflection and transmission as

R(ω) = |g(ω)|2, T (ω) = |1 + g(ω)|2

g(ω) = iγ1(ω2 − iη2 − ω) + γ2(ω1 − iη1 − ω)− 2γ12κ

(ω1 − iγ1 − iη1 − ω)(ω2 − iγ2 − iη2 − ω)− (κ− iγ12)2.

(S.13)

When η1,2 is small, the expression reduces back to equation (S.10). When η1,2 is substantial,neither the reflectivity peaks nor the transmission dips follow the eigenvalues Ω1,2.

To illustrate this point, we consider a system tuned to accidental degeneracy (ω1 = ω2 ≡ ω0)with only one resonance radiating (γ1 = 1, γ2 = γ12 = 0) and with equal gain/loss for the tworesonances (η1 = η2 ≡ η0). This system has linear Dirac dispersion Ω1,2 = ω0 ± κ for theeigenfrequency of the Hermitian part, while the complex eigenvalues exhibit an exceptionalpoint at κ = 1/2. In Fig. S7, we plot the reflection and transmission spectrum for a system withsubstantial non-radiative loss (η0 = 0.45, Fig. S7a), a purely passive system (η0 = 0, Fig. S7b),and a system with substantial gain (η0 = −0.25, Fig. S7c). On top of the plots, we indicate thereflection peaks (blue solid lines) and transmission dips (red solid lines), as well as the eigen-frequency of the Hermitian part (green dashed lines) and the real part of the eigenvalues (greensolid lines). As can be seen, in the passive case the reflection peaks and transmission dips followthe linear dispersion of Ω1,2, whereas in the cases with substantial non-radiative loss or materialgain, the reflection peaks and transmission dips show branching behavior instead.

Section IX, Full derivation using temporal coupled-mode theory (TCMT)Here, we provide the derivation of reflectivity using temporal coupled-mode theory (TCMT) [5–10] in a more general setup than the previous two sections. We will account for the direct(non-resonant) reflection process in between layers of dielectric, include arbitrary number ofn resonances (as there are other nearby resonances that modify the spectrum) with arbitrarysymmetries, and account for the non-radiative loss. We use the resulting expression to fit theexperimentally measured reflection data and extract the Hamiltonian matrix elements and eigen-values.

The time evolution of these n resonances, whose complex amplitudes are denoted by ann× 1 column vector A, is described by the Hamiltonian H and a driving term,

dA

dt= −iHA+KTs+, (S.14)

where the Hamiltonian is an n× n non-Hermitian matrix

H = Ω− iΓ− iγnr, (S.15)

with Ω denoting its Hermitian part that describes a lossless system, −iΓ denoting its anti-Hermitian part from radiation loss, and −iγnr its anti-Hermitian part from non-radiative decays.The non-radiative decay includes both material absorption and scattering from surface rough-ness (in our system, surface roughness is the primary contribution [2]), and here we consider itto be the same for all resonances, so γnr is a real number instead of a matrix.



Reflectivity measurements couple the n resonances to the incoming and outgoing planewaves,whose complex amplitudes we denote by two 2 × 1 column vectors, s+ and s−. The direct re-flection and transmission of the planewaves through the slab (in the absence of resonances) aredescribed by a 2× 2 complex symmetric matrix C, and

s− = Cs+ +DA, (S.16)

where D and K in equation (S.14) are 2 × n complex matrices denoting coupling between theresonances and the planewaves. We approximate the direct scattering matrix C by that of ahomogeneous slab whose permittivity is equal to the spatial average of the PhC slab [8–10].Lastly, outgoing planewaves into the silica substrate are reflected at the silica-silicon interface,so

s2+ = e2iβhsr23s2−, (S.17)

where hs is the thickness of the silica substrate with refractive index ns = 1.46, β =√

n2sω

2

c2 − |k∥|2is the propagation constant in silica, and r23 is the Fresnel reflection coefficient between silicaand the underlying silicon. The formalism described above is the same as Ref [9] except thathere we account for the coupling between the n resonances (off-diagonal terms of H).

For steady state with e−iωt time dependence, we solve for vector A from equation (S.14) toget the scattering matrix of the whole system that includes both direct and resonant processes,

s− = (C + Cres)s+, (S.18)

where the effect of the n resonances is captured in a 2× 2 matrix

Cres = iD(ω −H)−1KT. (S.19)

We can solve equation (S.17) and equation (S.18) to obtain the reflectivity

R(ω) =

∣∣∣∣s1−s1+

∣∣∣∣2

. (S.20)

In this expression, the only unknown is Cres. Therefore, by comparing the experimentally mea-sured reflectivity spectrum and the one given by TCMT in equation (S.20), we can extract theunknown parameters in the resonant scattering matrix Cres and obtain the eigenvalues of theHamiltonian H .

The remaining task is to write Cres using as few unknowns as possible so that the eigenvaluesof H can be extracted unambiguously. In equation (S.19), there are a large number of unknownsin the matrix elements of H , D, and K, but there is much redundancy because the matrixelements are not independent variables and because Cres is independent of the basis choice.Below, we show that we can express Cres with only 2n + 1 unknown real numbers, and these2n+ 1 real numbers are enough to determine the n complex eigenvalues of H .

First, we normalize the amplitudes of A and s± such that their magnitudes squared are theenergy of the resonances per unit cell and the power of the incoming/outgoing planewaves per



unit cell, respectively. Then, energy conservation, time-reversal symmetry, and C2 rotationalsymmetry of the PhC slab [5,13] require the direct scattering matrix to satisfy C† = C∗ = C−1

and the coupling matrices to satisfy D†D = 2Γ, K = D, and CD∗ = −D. It follows that thematrix Γ is real and symmetric. Next, using the Woodbury matrix identity and these constraints,we can rewrite equation (S.19) as

Cres = −2W (2 +W )−1 C, (S.21)

where W ≡ iD(ω − Ω+ iγ0)−1D† is a 2-by-2 matrix.We note that the matrix W , and therefore the matrix Cres, is invariant under a change of

basis for the resonances through any orthogonal matrix U (where Ω is transformed to UΩU−1,and D is transformed to DU−1). The eigenvalues of H is also invariant under a change ofbasis. Therefore, we are free to choose any basis. We kept the basis choice general in the two-resonance system described in the previous two sections, but here for the purpose of fitting, weneed as few unknowns as possible, and it is most convenient to choose the basis where Ω isdiagonal. So we let Ωij = Ωjδij , with Ωjnj=1 being the eigenvalues of Ω.

To proceed further, we note that the PhC slab sits on a silica substrate with ns = 1.46 and isimmersed in a liquid with n = 1.48, so the structure is nearly symmetric in the z direction. Themirror symmetry requires the coupling to the two sides to be symmetric or anti-symmetric [7],

D1j

D2j≡ σj = ±1, j = 1, . . . , n, (S.22)

where σj = 1 for TE-like resonances and σj = −1 for TM-like resonances, in the conventionwhere (Ex, Ey) determines the phase of Aj and s±. Then, the diagonal elements of Γ are relatedto D by Γjj ≡ γj = |D1j|2, and in this basis we have

W =n∑

j=1

iγjω − Ωj + iγnr

(1 σj

σj 1

). (S.23)

This completes our derivation. Equations (S.21) and (S.23) provide an expression for Cres thatdepends only on 2n+ 1 unknown non-negative real numbers: the n eigenvalues Ωjnj=1 of theHermitian matrix Ω, the n diagonal elements γjnj=1 of the real-symmetric radiation matrix Γin the basis where Ω is diagonal, and the non-radiative decay rate γnr.

At each angle and each polarization, we fit the experimentally measured reflectivity spec-trum to the TCMT expression equation (S.20) to determine these 2n + 1 unknown parameters.Fig. 3a and Fig. S8a show the comparison between the experimental reflectivity spectrum andthe fitted TCMT reflectivity spectrum at some representative angles. The near-perfect agree-ment between the two demonstrates the validity of the TCMT model.

To obtain the eigenvalues of the Hamiltonian H , we also need to know the off-diagonalelements of Γ. From D†D = 2Γ and D1j/D2j = σj , we see that Γij = 0 when resonance i andresonance j have different symmetries in z (i.e. when σiσj = 1), and that Γij = ±√

γiγj when



σiσj = 1. In the latter case, the sign of Γij depends on the choice of basis; the eigenvalues ofH are independent of the basis choice, so to calculate the eigenvalues of H , we can simply takethe positive roots of all of the non-zero off-diagonal elements of Γ.

We note that the model Hamiltonians introduced previously, such as equation (1) in the maintext and equation (S.6) above, are all special cases of the general Hamiltonian in equation (S.15)that we consider in the TCMT formalism in this section. The TCMT formalism in this sectiondoes not make assumptions on the particular forms of the matrix elements (aside from basicprinciples such as energy conservation and time-reversal symmetry), so it can be used as anunbiased method for analyzing the experimental data.



References[1] Sakoda, K. Proof of the universality of mode symmetries in creating photonic dirac cones.

Opt. Express 20, 25181–25194 (2012).

[2] Lee, J. et al. Observation and differentiation of unique high-Q optical resonances nearzero wave vector in macroscopic photonic crystal slabs. Phys. Rev. Lett. 109, 067401(2012).

[3] Rotter, I. A non-Hermitian Hamilton operator and the physics of open quantum systems.J. Phys. A 42, 153001 (2009).

[4] Dembowski, C. et al. Experimental observation of the topological structure of exceptionalpoints. Phys. Rev. Lett. 86, 787–790 (2001).

[5] Suh, W., Wang, Z. & Fan, S. Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities. IEEE J. Quantum Electron. 40, 1511–1518 (2004).

[6] Haus, H. A. Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ,1984).

[7] Joannopoulos, J. D., Johnson, S. G., Winn, J. N. & Meade, R. D. Photonic Crystals:Molding the Flow of Light (Princeton University Press, 2008), 2 edn.

[8] Fan, S. & Joannopoulos, J. D. Analysis of guided resonances in photonic crystal slabs.Phys. Rev. B 65, 235112 (2002).

[9] Fan, S., Suh, W. & Joannopoulos, J. D. Temporal coupled-mode theory for the Fanoresonance in optical resonators. J. Opt. Soc. Am. A 20, 569–572 (2003).

[10] Hsu, C. W. et al. Observation of trapped light within the radiation continuum. Nature 499,188–191 (2013).

[11] Miroshnichenko, A. E., Flach, S. & Kivshar, Y. S. Fano resonances in nanoscale structures.Rev. Mod. Phys. 82, 2257 (2010).

[12] Hsu, C. W., DeLacy, B. G., Johnson, S. G., Joannopoulos, J. D. & Soljacic, M. Theoreticalcriteria for scattering dark states in nanostructured particles. Nano Lett. 14, 2783–2788(2014).

[13] Hsu, C. W. et al. Bloch surface eigenstates within the radiation continuum. Light: Science& Applications 2, e84 (2013).



SUPPLEMENTARY FIGURES

Freq

uenc

y ω

a/2π

c

0.54

0.55

0 0.01

X

kxa/2π

Γ

xy

z2r

a

Re(Hz) Im(Hz)

Re(Hz) Im(Hz)

∼ ψ1 + ψ2

∼ ψ1 − ψ2

ψ1 Re(Hz)

ψ2

Im(Hz)

Hz (a.u.)

+_ 0

Hermitian non-Hermitiana)

Re(Hz) Im(Hz)

Re(Hz) Im(Hz)

ψ1

Freq

uenc

y R

e(ω

)a/2

πc

0.585

0.605

0 0.01

X

kxa/2π

Γ

Re(Hz)

z = 0

z = h

z = h/2

z = h/2

z = h/2

ψ2

Im(Hz)

∼ ψ1 − iψ2

∼ ψ1 − iψ2

Basis

Hz (a.u.)

+_ 0

b)

Basis

Supplementary Figure 1: Eigenfunctions in the Hermitian PhC and the non-HermitianPhC slab. a, Numerically calculated band structure and eigenfunctions for the Hermitian 2DPhC. The basis eigenfunctions at the Γ point (Ψ1 and Ψ2) are shown in the black box. Theeigenfunctions away from Γ point (at kxa/2π = 0.0035) are shown in the red and blue boxes,agreeing well with the analytical prediction of Ψ1 ± Ψ2. b, Corresponding results for the non-Hermitian PhC slab, showing the basis eigenfunctions at the Γ point (black box) and the twoeigenfunctions near the exceptional point (kxa/2π = 0.0035, red and blue boxes). The twoeigenfunction near EP almost coalesce and both agree with the analytical prediction of Ψ1−iΨ2.



b)a) c)r = rcr < rc r > rc

-3

0*10-3

0.60

0.59Re(

ω) a

/2πc

Im(ω

) a/2

πc

|k|a/2π0

-1

-2

-4

0.61

Γ KM

0.01 0.01|k|a/2π

00.01 0.01|k|a/2π

00.01 0.01

Γ KM Γ KM

Supplementary Figure 2: Simulation results of the complex eigenvalues of the PhC slabswith and without accidental degeneracy. The real (upper panels) and imaginary (lower pan-els) parts of the complex eigenvalues are shown for structures with (b) and without (a, c) acci-dental degeneracy. The bands with quadrapole modes in the middle of the Brillouin zone areshown in red solid lines, while the bands with the dipole mode are shown in blue solid linesand gray dashed lines. The bands shown in red and blue solid lines couple to each other, whilethe band in gray dashed lines is decoupled from the other two due to symmetry. When acci-dental degeneracy happens (at r = rc as shown in b), the characteristic branching features areobserved, demonstrating the existence of EPs. When the accidental degeneracy is lifted, thequadrupole band splits from the dipole bands from different directions: from the bottom whenradius of the holes is too small (r < rc as shown in a), or from the top when the radius is toobig (r > rc as shown in c).



e)

h

Re(

ω) a

/2πc

0 0.006

A’ B’ C’ D’ A’

Eigenvalues switch when looping around EPsf)

Im(ω

) a/2

πc

Re(ω) a/2πc0.594 0.6

0*10-3

-1

-2

-3

d)

kxa/2π

r (nm)

r1=104.5

r2

A’

D’

B’

C’

Ep

kya/2π

r3=101.50.002 0.006

kxa/2π0.002 0.002 0.006 0.006

r = r1 r = r1 r = r1r = r3 r = r3

0.6

b)

h

Re(

ω) a

/2πc

0 0.01

A B C D A

Eigenvalues stay when not looping around EPs

Im(ω

) a/2

πc

c)Re(ω) a/2πc

0.596 0.602*10-3

0

-1

-2

-3

a)

r (nm)

r2

A

D

B

C

Ep

kya/2πkxa/2π

r1=104.5

r3=101.50.006 0.01

kxa/2π0.006 0.006 0.01 0.01

r = r1 r = r1 r = r1r = r3 r = r3

0.6

g)Re(ω) a/2πc

Im(ω

) a/2

πc

0.593 0.6010*10-3

-1

-2

-3

r (nm)

r1=104.5

r2”A”

D”

B”

C”

Ep’’

Γ

r1=101.5

h)

kx

ky

r2’’’A’’’

D’’’

B’’’

C’’’Ep’’’

Γ

r1=104.5

r1=101.5

r (nm) Re(ω) a/2πc

Im(ω

) a/2

πc

0.593 0.6010*10-3

-1

-2

-3

Existence of EP in every direction

kx

ky

Γ Γ

Supplementary Figure 3: Existence of EP along every direction in the momentum spacefor the realistic PhC slab structure. a, A loop is created in the parameter space of the structure(A → B → C → D → A), which does not enclose the EP of the system (point Ep). Here,r is the radius of the air holes, and kx,y are the in-plane wavevectors. b, For each point alongthe loop, we numerically calculate the eigenvalues of the PhC slab with the corresponding holeradius at the corresponding in-plane wavevector. c, The complex eigenvalues return to their ini-



-tial positions at the end of the loop (namely, the blue dot and the red dot come back to them-selves) when the system parameters come back to point A. d,e,f, Another loop is created(A′ → B′ → C ′ → D′ → A′), which encloses an EP of the system (the same point Ep asin a). Following this new loop, the two eigenvalues switch their positions at the end of the loop(namely, the blue dot and the red dot switch their positions) when the system parameters comeback to point A′. g,h, The two complex eigenvalues always switch their positions when wechoose the right loops along other directions in the momentum space (Γ to N in g and Γ to Min h).

Reflectivity spectrum

ΓM Xs-polarizedp-polarized

Wav

elen

gth

(nm

)

569

567

563

561

559ΓM Xp-polarizeds-polarized

Angle (degrees)2 0 21 1

Angle (degrees)2 0 21 1

1

0

Supplementary Figure 4: Experimental results of reflectivity showing an accidental Diraccone. Light with different polarizations (s and p) is selected to excite different resonances of thePhC slab along different directions in the k space. Depending on the choice of polarization, thetwo bands forming the conical dispersion are excited (s-polarized along Γ-X and p-polarizedalong Γ-M, shown in the left panel), or the flat band in the middle is excited (p-polarized alongΓ-X and s-polarized along Γ-M, shown in the right panel).



Angle (degrees)0 10.5 1.5 2



r = r’

567

564

561

558

570

567

579

573

570

576567

564

561

558

570

Wav

elen

gth

(nm

)

r > r’ r < r’

0

1a) b) c)

Supplementary Figure 5: Experimental results of reflectivity from PhC slabs with andwithout accidental degeneracy. Angle-resolved reflectivity along the Γ to X direction, mea-sured for three different PhC slabs: a, with bigger hole radius than the structure with accidentaldegeneracy (r > r′); b, with accidental degeneracy (r = r′); c, with smaller hole radius thanthe structure with accidental degeneracy (r < r′). The reflectivity peaks of the structures with-out accidental degeneracy (a,c) follow quadratic dispersions; while the reflectivity peaks of thestructure with accidental degeneracy (b) follow linear Dirac dispersion. Data shown in b is thesame data as in Fig. 2c and the left panel of Fig. S3.



a)

0

1

Re(ω) a.u.)

ω

Im(ω)−

Rresonance

b) Reflection from two coupled resonances

0

1

R

Im(ω)

Re(ω) a.u.)

Re(ω) a.u.)0

1

R

Im(ω)EP

Re(ω) a.u.)0

1

R

Im(ω)

CRIT 1.2 b

0.8 b

b b

0.3 b

2 b

(a.u.)

(a.u.)

(a.u.)

Reflection from one resonance

(a.u.)

Supplementary Figure 6: Illustrative reflectivity spectrum from one resonance and fromtwo coupled resonances. a, When a single resonance dominates, the reflectivity peak is at thesame position as the real part of the complex eigenvalue. b, With two coupled resonances, thereflectivity peaks (red arrows) no longer follow the eigenvalues of the system (blue circles). Aswe vary the radiation loss γ1,2 of the two resonances while fixing the eigenvalues Ω1,2 of theHermitian matrix, we see the complex eigenvalues vary, whereas the reflectivity peaks are fixed.The middle panel of (b) shows a situation when the two complex eigenvalues coalesce into anEP.



passive with gainwith lossa) b) c)

ω -

ω a

.u.)

ω -

ω a.

u.)

a.u.)

-22

0-2

20

-2 20 -2 20 -2 20

0 1 0 1 0 2 a.u.) a.u.)

& above

Reflection

Transmission

Reflection

Transmission

Reflection

Transmission

Supplementary Figure 7: Effect of material loss and gain on the reflection and transmis-sion spectra. Reflection spectra (upper panels) and transmission spectra (lower panels) areplotted for three systems with the same radiation losses, but different material gain/loss: withmaterial loss (a), purely passive with no gain or loss (b), and with material gain (c). For thepurely passive system (b), the reflection peak (blue solid line) and the transmission dip (redsolid line) correspond to the eigenvalue of the Hermitian part of the Hamiltonian (green dashedline), and show the linear Dirac dispersion. When material loss (a) or gain (c) is added to thesystem, the reflection peaks and transmission dips deviate from the linear Dirac dispersion andshow branching behavior instead. The system parameters are given in Section VIII.



a) Experiment Coupled-Mode Theory

0

1R

0

10

10

10

1

R

R

R

R

nm)550 580 nm)550 580

= 1.00

= 0.5

= 0.24

= 0.14

= 0

Angle (degrees)

wav

elen

gth

(nm

)

1.5 1 0 1 1.50.5 0.5

ΓM X

570

560

562

564

566

568

0.6

0.598

0.596

0.594

0.592

a/2π

cAngle (degrees)

1.5 1 0 1 1.50.5 0.5

0

-1

-0.5

-1.5

-2

Fitting results before diagonalization

−a/

2πc

ΓM X

b)

0.59

Supplementary Figure 8: TCMT fitting and visualization of accidental Dirac cone. a, Ex-amples of reflection spectrum measured at five different incident angles (0, 0.14, 0.24, 0.5and 1) along the Γ-X direction for s polarization, with comparison to the TCMT expression inequation (S.20) after fitting. Dotted lines indicate the resonances, with the two relevant reso-nances marked in red and blue. b, Parameters obtained from the TCMT fitting. The eigenvaluesfor the Hermitian part of the Hamiltonian, Ω1,2, are shown in the left panel and reveal the Diracdispersion arising from accidental degeneracy. The diagonal terms for the anti-Hermitian partof the Hamiltonian, γ1,2, are shown in the right panel. Note that the anti-Hermitian part ofthe Hamiltonian also has off-diagonal terms, so Ω1,2 and γ1,2 are not the eigenvalues of theHamiltonian. The eigenvalues of the Hamiltonian are shown in Fig. 3 of the main text.



dirac cones - jdj.mit.edu

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